Galerkin finite element solution for Benjamin–Bona–Mahony–Burgers equation with cubic B-splines

Abstract

In this article, we study solitary-wave solutions of the nonlinear Benjamin–Bona–Mahony–Burgers(BBM–Burgers) equation based on a lumped Galerkin technique using cubic B- spline finite elements for the spatial approximation. The existence and uniqueness of solutions of the Galerkin version of the solutions have been established. An accuracy analysis of the Galerkin finite element scheme for the spatial approximation has been well studied. The proposed scheme is carried out for four test problems including dispersion of single solitary wave, interaction of two, three solitary waves and development of an undular bore. Then we propose a full discrete scheme for the resulting IVP. Von Neumann theory is used to establish stability analysis of the full discrete numerical algorithm. To display applicability and durableness of the new scheme, error norms L2, L∞ and three invariants I1,I2 and I3 are computed and the acquired results are demonstrated both numerically and graphically. The obtained results specify that our new scheme ensures an apparent and an operative mathematical instrument for solving nonlinear evolution equation.